Beyond the Projection Postulate and Back: Quantum Theories with Generalised State-Update Rules
Vincenzo Fiorentino, Stefan Weigert
TL;DR
The paper presents a systematic framework for generalized state-update rules (GURs) to explore alternatives to the Lüders projection postulate. It defines core operational requirements (A1–A6) for updating rules on both single and composite systems and constructs several foil theories (e.g., Locally-Lüders, Passive Quantum Theory, Depolarising, and Probability-amplifying updates) to contrast with standard quantum theory. It shows that coherence suffices to derive Lüders updates for non-composite systems, but both coherence and composition compatibility are needed to extend uniquely to composite systems, thereby singling out $\,\omega^{\mathsf{L}}_{AB}$. The framework clarifies how alternative update rules can maintain no-signalling while altering entanglement correlations, preparation indistinguishability, or local tomography, and it discusses implications for reconstructing quantum theory and for future work on nonlocality and information-disturbance trade-offs in generalized theories.
Abstract
Are there consistent and physically reasonable alternatives to the projection postulate? Does it have unique properties compared with acceptable alternatives? We answer these questions by systematically investigating hypothetical state-update rules for quantum systems that nature could have chosen over the Lüders rule. Among other basic properties, any prospective rule must define unique post-measurement states and not allow for superluminal signalling. Particular attention will be paid to consistently defining post-measurement states when performing local measurements in composite systems. Explicit examples of valid unconventional update rules are presented, each resulting in a distinct, well-defined foil of quantum theory. This framework of state-update rules allows us to identify operational properties that distinguish the projective update rule from all others and to put earlier derivations of the projection postulate into perspective.